Tachath means under, beneath, below, or instead of. It appears about 505 times. It expresses spatial position (underneath) and substitution (in place of).
The substitutionary sense of tachath is theologically rich. The ram 'instead of' Isaac (Genesis 22:13), the Levites 'in place of' the firstborn (Numbers 3:12), and ultimately Christ who died 'in our place' use this concept. The preposition captures the substitutionary heart of atonement.